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either the capacitor needs to be replaced or the control board is not sending power to the fan motor consistently.
I've had the voltmeter connected to the wire to the motor for a number of days and the voltage to the motor is always around 120V +/- .5V when the relay is kicked on and the Voltage is 0V when the relay turns off. It sounds like the capacitor has gone weak and isn't "kicking" the motor on first time in the morning when the unit is cold?
Yes I would advise you to replace the capacitor.